1. Reliability, Redundancy, and Resiliency
Review of probability theory
Component reliability
Confidence
Redundancy
Reliability diagrams
Intercorrelated Failures
System resiliency
Resiliency in fixed fleets
Add in reliability trees
© David L. Akin - All rights reserved

2. Review of Probability Probability that A occurs
Probability that A does not occur
Sum of all probable outcomes

3. Review of Probability Probability of both A and B occurring
Probability of either A or B occurring

4. Utility Theory
Probability of an outcome does not determine utility of the outcome
Use probability and utility to determine expected value of outcome

5. Utility Example
Maryland State Lottery - pick six numbers out of 49 (any order)
Assume $10,000,000 jackpot
lame - tie into aerospace!
Discussion of risk adversity - also lame

6. Component Reliability
Burn-in Failures
Operating Failures
End-of-life Failures
Failure Rate l
Fix so you can see it on the projector
Told AP101 stories
Time

7. Reliability Analysis
Failure rate is defined as fraction of currently operating units failing per unit time
The trend of operating units with time is then

8. Reliability Analysis (continued)
Evaluation of the definite integrals gives
Assuming that l is constant over the operating lifetime,
At t=1/ l, 1/e of the original units are still operating (defined as mean time between failures)

9. Reliability Analysis (continued)
Frequently assess component reliability based on reciprocal of failure rate l : where MTBF=mean time between failures
For a mission duration of N hours, estimate of component reliability becomes

10. Verifying a Reliability Estimate
Given a unit reliability of R, what is the probability P of testing it 20 times without a failure?
What is the probability Q that you will see one or more failures?
R= P= Q=.1821
R= P= Q=.6416
R= P= Q=.8784

11. Confidence
The confidence C in a test result is equal to the probability that you should have seen worse results than you did P(observed and better outcomes) + C =1

12. Example of Confidence 100 vehicle flights with 1 failure
Assume a reliability value of R
Trade off reliability with confidence values
Chenge 99 to 100; fix plot (still for 95 flights from last year)

13. Definition of Redundancy
Probability of k out of n units working = (number of permutations of k out of n) x P(k units work) x P(n-k units fail)

14. Redundancy Example 3 parallel computers, each has reliability of 95%:
Probability all three work
Probability exactly two work
Probability exactly one works
Probability that none work

15. Redundancy Example 3 parallel computers, each has reliability of 95%:
Probability all three work
Probability at least two work
Probability at least one works
Probability that none work

16. Reliability Diagrams Example of Apollo Lunar Module ascent engine
Three valves in each of oxidizer and fuel lines
One in each set of three must work
Rv=0.9 --> Rsystem=.998 Rv Rv Rv Rv Rv Rv

17. Reliability Diagrams (how not to…)Rv Rv Rv Rv
Rv=0.9 --> Rsystem=.998 Rv Rv Rv Rv Rv Rv
Rv=0.9 --> Rsystem=.993 Rv Rv

18. Intercorrelated Failures
Some failures in redundant systems are common to all units
Software failures
“Daisy-chain” failures
Design defects
Following a failure, there is a probability f that the failure causes a total system failure

19. Intercorrelated Failure Example
3 parallel computers, each has reliability of 95%, and a 30% intercorrelated failure rate:
Probability all three work
Probability exactly two work (one failure)
Probability the failure is benign (system works)
Probability of intercorrelated failure (system dies)

20. Intercorrelated Failure Example
(continued from previous slide)
Probability exactly one works (2 failures)
Probability that both failures are benign
Probability that a failure is intercorrelated

21. Redundancy Example with Intercorrelation
3 parallel computers, each has reliability of 95%, and a 30% intercorrelated failure rate:
Probability all three work
Probability at least two work
Probability at least one works

22. System Reliability with 30% Intercorrelation

23. Concept of System Resiliency
Initial flight schedule ( ( ( ( ( ( ( ( ( ( (
Hiatus period following a failure ( @ ( ( ( ( (
Backlog of payloads not flown in hiatus ( ( ( (
Surge to fly off backlog ( @ ( ( (( (( (((
Resilient if backlog is cleared before next failure occurs (on average)
Think about transistion effects here

24. Resiliency Variables r - nominal flight rate, flts/yr
d - down time following failure (yrs)
k - fraction of flights in backlog retained
S - surge flight rate/nominal flight rate
m - average/expected flights between failures
rd - number of missed flights
krd - number of flights in backlog
(S-1)r - backlog flight rate
Take out transistion effects!

25. Definition of Resiliency
Example for Delta launch vehicle
r = 12 flts/yr
d = 0.5 yrs
k = 0.8
S = 1.5
m = 30
Srkd/(S-1) = 14.4 < 30 - system is resilient!

26. Shuttle Resiliency r = 9 flts/yr d = 2.5 yrs k = 0.8
S = .67 (6 flts/yr)
m = 25
System has negative surge capacity due to reduction in fleet size - cannot ever recover from hiatus without more extreme measures

27. Modified Resiliency
k’ - retention rate of all future payloads (k’≤S for S<1)
New governing equation for resiliency:
Implication for shuttle case:
k<.417 to achieve modified resiliency
Finished at 11:48(!!)